Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf\n metrics on moduli spaces

2015/08/09 by Jinsung Park, Leon A. Takhtajan, Park, Jinsung +3
Mathematics · #32G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometry and complex manifolds #Primary 14H60 #Secondary 53C80

paper · pdf · doi:10.48550/arxiv.1508.02102

openalex publication_date 2015/08/09 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

For the TZ metric on the moduli space mathscrM_0,n of n-pointed\nrational curves, we construct a K "ahler potential in terms of the Fourier\ncoefficients of the Klein's Hauptmodul. We define the space\n mathfrakS_g,n as holomorphic fibration\n mathfrakS_g,n\→ mathfrakS_g over the Schottky space\n mathfrakS_g of compact Riemann surfaces of genus g, where the fibers\nare configuration spaces of n points. For the tautological line bundles\n mathscrL_i over mathfrakS_g,n we define Hermitian metrics h_i\nin terms of Fourier coefficients of a covering map J of the Schottky domain.\nWe define the regularized classical Liouville action S and show that\n\exp S/\π is a Hermitian metric in the line bundle\n mathscrL=\⊗_i=1ⁿ mathscrL_i over mathfrakS_g,n. We\nexplicitly compute the Chern forms of these Hermitian line bundles\nc₁(
mathscrL_i,h_i)=
frac43
omega_
mathrmTZ,i,
quad\nc₁(
mathscrL,
exp
S/
pi
)=
frac1
pi²
omega_
mathrmWP. We\nprove that a smooth real-valued function - mathscrS=-S+\π\∑_i=1ⁿ\log\nh_i on mathfrakS_g,n, a potential for this special difference of WP\nand TZ metrics, coincides with the renormalized hyperbolic volume of a\ncorresponding Schottky 3-manifold. We extend these results to the\nquasi-Fuchsian groups of type (g,n).\n

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